Whisk water, egg, vanilla, and melted butter in a large bowl.
Add 2 cups flour, sugar, yeast, and salt. Mix into a sticky dough.
Gradually add remaining flour until dough forms a ball. Knead until smooth and elastic.
Shape into a ball, oil the top, and let rise (covered) for 45–60 mins until doubled.
Punch down dough, divide into 8–12 pieces, and roll into balls.
Place on parchment-lined sheet, cover, and let rise another 30–45 mins.
Make an indent in each ball, fill with jam.
Bake at 350°F (175°C) for 15–20 mins until golden and jam is bubbly.
Mix icing ingredients until smooth.
Drizzle icing over warm donuts and dust with powdered sugar if desired.